The Laplace transform on time scales revisited
Abstract
In this work, we reexamine the time scale Laplace transform as defined by Bohner and Peterson [M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001; M. Bohner, A. Peterson, Laplace transform and Z-transform: Unification and extension, Methods Appl. Anal. 9 (1) (2002) 155-162]. In particular, we give conditions on the class of functions which have a transform, develop an inversion formula for the transform, and further, we provide a convolution for the transform. The notion of convolution leads to considering its algebraic structure--in particular the existence of an identity element--motivating the development of the Dirac delta functional on time scales. Applications and examples of these concepts are given.
- Publication:
-
Journal of Mathematical Analysis and Applications
- Pub Date:
- August 2007
- Bibcode:
- 2007JMAA..332.1291D
- Keywords:
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- Time scale;
- Laplace transform;
- Convolution;
- Dirac delta